Theorem pnt 21261

Description: The Prime Number Theorem: the number of prime numbers less than x tends asymptotically to x / log ( x ) as x goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)

Assertion

Ref	Expression
pnt	|- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Proof of Theorem pnt

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 9046	. . . . . . 7 |- 1 e. RR
2	1	rexri 9093	. . . . . 6 |- 1 e. RR*
3		1lt2 10098	. . . . . 6 |- 1 < 2
4		df-ioo 10876	. . . . . . 7 |- (,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x < z /\ z < y ) } )
5		df-ico 10878	. . . . . . 7 |- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
6		xrltletr 10703	. . . . . . 7 |- ( ( 1 e. RR* /\ 2 e. RR* /\ w e. RR* ) -> ( ( 1 < 2 /\ 2 <_ w ) -> 1 < w ) )
7	4, 5, 6	ixxss1 10890	. . . . . 6 |- ( ( 1 e. RR* /\ 1 < 2 ) -> ( 2 [,) +oo ) C_ ( 1 (,) +oo ) )
8	2, 3, 7	mp2an 654	. . . . 5 |- ( 2 [,) +oo ) C_ ( 1 (,) +oo )
9		resmpt 5150	. . . . 5 |- ( ( 2 [,) +oo ) C_ ( 1 (,) +oo ) -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
10	8, 9	mp1i 12	. . . 4 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) )
11	8	sseli 3304	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. ( 1 (,) +oo ) )
12		ioossre 10928	. . . . . . . . . . 11 |- ( 1 (,) +oo ) C_ RR
13	12	sseli 3304	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> x e. RR )
14	11, 13	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x e. RR )
15		2re 10025	. . . . . . . . . . 11 |- 2 e. RR
16		pnfxr 10669	. . . . . . . . . . 11 |- +oo e. RR*
17		elico2 10930	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ +oo e. RR* ) -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) ) )
18	15, 16, 17	mp2an 654	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x /\ x < +oo ) )
19	18	simp2bi 973	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> 2 <_ x )
20		chtrpcl 20911	. . . . . . . . 9 |- ( ( x e. RR /\ 2 <_ x ) -> ( theta ` x ) e. RR+ )
21	14, 19, 20	syl2anc 643	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. RR+ )
22		0re 9047	. . . . . . . . . . . 12 |- 0 e. RR
23	22	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 e. RR )
24	1	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 e. RR )
25		0lt1 9506	. . . . . . . . . . . 12 |- 0 < 1
26	25	a1i 11	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 0 < 1 )
27		eliooord 10926	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( 1 < x /\ x < +oo ) )
28	27	simpld 446	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> 1 < x )
29	23, 24, 13, 26, 28	lttrd 9187	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> 0 < x )
30	13, 29	elrpd 10602	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> x e. RR+ )
31	11, 30	syl 16	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> x e. RR+ )
32	21, 31	rpdivcld 10621	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / x ) e. RR+ )
33	32	adantl 453	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / x ) e. RR+ )
34		ppinncl 20910	. . . . . . . . . . 11 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
35	14, 19, 34	syl2anc 643	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. NN )
36	35	nnrpd 10603	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. RR+ )
37	13, 28	rplogcld 20477	. . . . . . . . . 10 |- ( x e. ( 1 (,) +oo ) -> ( log ` x ) e. RR+ )
38	11, 37	syl 16	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. RR+ )
39	36, 38	rpmulcld 10620	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. RR+ )
40	21, 39	rpdivcld 10621	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
41	40	adantl 453	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) e. RR+ )
42	31	ssriv 3312	. . . . . . . 8 |- ( 2 [,) +oo ) C_ RR+
43		resmpt 5150	. . . . . . . 8 |- ( ( 2 [,) +oo ) C_ RR+ -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) )
44	42, 43	ax-mp 8	. . . . . . 7 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) )
45		pnt2 21260	. . . . . . . 8 |- ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1
46		rlimres 12307	. . . . . . . 8 |- ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) ~~> r 1 -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
47	45, 46	mp1i 12	. . . . . . 7 |- ( T. -> ( ( x e. RR+ |-> ( ( theta ` x ) / x ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
48	44, 47	syl5eqbrr 4206	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / x ) ) ~~> r 1 )
49		chtppilim 21122	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1
50	49	a1i 11	. . . . . 6 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ~~> r 1 )
51		ax-1ne0 9015	. . . . . . 7 |- 1 =/= 0
52	51	a1i 11	. . . . . 6 |- ( T. -> 1 =/= 0 )
53	40	rpne0d 10609	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
54	53	adantl 453	. . . . . 6 |- ( ( T. /\ x e. ( 2 [,) +oo ) ) -> ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) =/= 0 )
55	33, 41, 48, 50, 52, 54	rlimdiv 12394	. . . . 5 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) ~~> r ( 1 / 1 ) )
56	14	recnd 9070	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> x e. CC )
57		chtcl 20845	. . . . . . . . . . . . 13 |- ( x e. RR -> ( theta ` x ) e. RR )
58	13, 57	syl 16	. . . . . . . . . . . 12 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. RR )
59	58	recnd 9070	. . . . . . . . . . 11 |- ( x e. ( 1 (,) +oo ) -> ( theta ` x ) e. CC )
60	11, 59	syl 16	. . . . . . . . . 10 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) e. CC )
61	56, 60	mulcomd 9065	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( x x. ( theta ` x ) ) = ( ( theta ` x ) x. x ) )
62	61	oveq2d 6056	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) )
63	39	rpcnd 10606	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) e. CC )
64	31	rpne0d 10609	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> x =/= 0 )
65	21	rpne0d 10609	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) -> ( theta ` x ) =/= 0 )
66	63, 56, 60, 64, 65	divcan5d 9772	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( ( theta ` x ) x. x ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
67	62, 66	eqtrd 2436	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
68	39	rpne0d 10609	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) x. ( log ` x ) ) =/= 0 )
69	60, 56, 60, 63, 64, 68, 65	divdivdivd 9793	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( ( ( theta ` x ) x. ( (π ` x ) x. ( log ` x ) ) ) / ( x x. ( theta ` x ) ) ) )
70	35	nncnd 9972	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> (π ` x ) e. CC )
71	38	rpcnd 10606	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) e. CC )
72	38	rpne0d 10609	. . . . . . . 8 |- ( x e. ( 2 [,) +oo ) -> ( log ` x ) =/= 0 )
73	70, 56, 71, 64, 72	divdiv2d 9778	. . . . . . 7 |- ( x e. ( 2 [,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) = ( ( (π ` x ) x. ( log ` x ) ) / x ) )
74	67, 69, 73	3eqtr4d 2446	. . . . . 6 |- ( x e. ( 2 [,) +oo ) -> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) = ( (π ` x ) / ( x / ( log ` x ) ) ) )
75	74	mpteq2ia 4251	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( ( theta ` x ) / x ) / ( ( theta ` x ) / ( (π ` x ) x. ( log ` x ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
76		ax-1cn 9004	. . . . . 6 |- 1 e. CC
77	76	div1i 9698	. . . . 5 |- ( 1 / 1 ) = 1
78	55, 75, 77	3brtr3g 4203	. . . 4 |- ( T. -> ( x e. ( 2 [,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
79	10, 78	eqbrtrd 4192	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 )
80		ppicl 20867	. . . . . . . . . 10 |- ( x e. RR -> (π ` x ) e. NN0 )
81	13, 80	syl 16	. . . . . . . . 9 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. NN0 )
82	81	nn0red 10231	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> (π ` x ) e. RR )
83	30, 37	rpdivcld 10621	. . . . . . . 8 |- ( x e. ( 1 (,) +oo ) -> ( x / ( log ` x ) ) e. RR+ )
84	82, 83	rerpdivcld 10631	. . . . . . 7 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. RR )
85	84	recnd 9070	. . . . . 6 |- ( x e. ( 1 (,) +oo ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
86	85	adantl 453	. . . . 5 |- ( ( T. /\ x e. ( 1 (,) +oo ) ) -> ( (π ` x ) / ( x / ( log ` x ) ) ) e. CC )
87		eqid 2404	. . . . 5 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) = ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) )
88	86, 87	fmptd 5852	. . . 4 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) : ( 1 (,) +oo ) --> CC )
89	12	a1i 11	. . . 4 |- ( T. -> ( 1 (,) +oo ) C_ RR )
90	15	a1i 11	. . . 4 |- ( T. -> 2 e. RR )
91	88, 89, 90	rlimresb 12314	. . 3 |- ( T. -> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 <-> ( ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) |` ( 2 [,) +oo ) ) ~~> r 1 ) )
92	79, 91	mpbird 224	. 2 |- ( T. -> ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1 )
93	92	trud 1329	1 |- ( x e. ( 1 (,) +oo ) |-> ( (π ` x ) / ( x / ( log ` x ) ) ) ) ~~> r 1

Syntax hints:   <-> wb 177   /\ w3a 936   T. wtru 1322   = wceq 1649   e. wcel 1721   =/= wne 2567   C_ wss 3280   class class class wbr 4172   e. cmpt 4226   |` cres 4839   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   +oocpnf 9073   RR*cxr 9075   < clt 9076   <_ cle 9077   / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   RR+crp 10568   (,)cioo 10872   [,)cico 10874   ~~> r crli 12234   logclog 20405   thetaccht 20826  πcppi 20829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-em 20784  df-cht 20832  df-vma 20833  df-chp 20834  df-ppi 20835  df-mu 20836